where \(\hat{m}\)  is the estimated transmission antenna index, \(\hat{l}\)  is the estimated OAM state, \({\hat{x}_p}\)  is the estimated radiated symbol. When the OAM signal is transmitted from the \(m - th\)  transmission antenna, the \(m - th\)  receive antenna pair are assumed to be located at the OAM circle region. In this case, the power of OAM signal at the \(m - th\)  receive antennas are obviously larger than the power of OAM signal at other receive antenna pairs. Hence, the transmission antenna index can be estimated by the receive antenna index with the maximum receive power of OAM signal. Based on the phase gradient method [1]}, the OAM state is estimated by the receive antenna pair with the maximum power of OAM signal. Furthermore, the maximum likelihood method is used to demodulate radiated symbols from the receive antenna pair. Consequently, the OAM-SM demodulation process is decomposed by the following three steps:
\(1) \quad \hat{m} = \mathop {\arg \;max}\limits _{1 \leqslant m \leqslant M} \left\lbrace  {{{\left| {{{\left[ {\mathbf {y}} \right]}_{m,1}}} \right|}^2}} \right\rbrace ,\qquad \mathrm {(16)}\) 
\(2) \quad \hat{l} = \frac{{{\phi _{\hat{m}\hat{m},1}} - {\phi _{\hat{m}\hat{m},2}}}}{\beta },\qquad \mathrm {(17)}\) 
\(3) \quad {\hat{x}_p} = \mathop {\arg \;\min }\limits _{x \in \mathcal {X}} \left\lbrace  {{{\left| {{{\left[ {\mathbf {y}} \right]}_{\hat{m},1}} - h_{\hat{m}\hat{m},1}^{\hat{l}}{x_p}} \right|}^2}} \right\rbrace ,\qquad \mathrm {(18)}\) 
